I currently work with Cypress only for api testing (a decision made by the company). There's no right or wrong answer. If it's the tool you want to use and you are comfortable with it, then go for it.
That being said, take into account that Cypress is a framework designed to do end to end or even component testing and therefore, even when you do not have a UI, it will always use the browser to perform actions (this in the long term, and depending the amount of tests, can have an impact in the run time). Lots of things are designed to interact with ui components and you need to patch/wrap your way to perform more complex stuff.
I personally had some issues with nested requests, or when having to perform multiple actions with responses. Asynchrony in general (handled in a weird way by cypress in my opinion).
If I could choose, I would go with a framework designed for api testing just because it will be more flexible and probably will have more tools for you to use when dealing with some of the things I mentioned above.
Hope this helps and good luck!